FIGURE OF A GRAVITATING BODY, Q\0 



sities are equal, this force will be to the whole weight as 

 Y. 1^ or f of the ellipticity to the radius ; and the portion of 

 the inclination remaining to be compensated by the primi- 

 tive disturbing force will be f of the whole, so that the 

 ellipticity must be to the proportional disturbing force as 5 

 to 2. And if the density of the sea be to the mean density 

 of the earth as 1 to n, the disturbing force, produced by its 



3 

 attraction, will be to the ellipticitv as — to 1, and the pri- 

 on '■ 



3 

 mitive disturbing force as 1 — r— to ]. 



The heights of the solar and lunar tides in equilibrium Tides of a ho- 

 having been found equal to '8097 and 2'0l66 feet respec- .phoroid, and 

 tjvely, on the supposition of the density of the sea being o' he sea as it 

 . 1 , , , 1 . I , actually exists, 



mconsiderable, they must be mcreased to 2*024 and 5*042 



for an imaginary planet of uniform density ; but since n is 

 in reality about 5f , and — - nearly i, the elUpticity must be 

 to the primitive disturbing force only as 1 to |- or 9 to 8, 

 and the height of the sides in equilibrium '91 1 and 2'269 

 respectively, and the joint height 3*18 feet. And when the 

 surface assumes any other form than that which affords the 

 equilibrium, the force tending to restore that form is always 

 less by one ninth than it appears to be when the attraction 

 of the elevated parts is neglected. The theory of the tides 

 must therefore be very materially modified by these consi- 

 derations, although they do not affect the general method 

 of explaining the phenomena. 



These calculations are also immediately applicable to the Ellipticity fronn 

 figure of an oblate spheroid : for it may easily be shown, ^^"'" nd*den- 

 that the difference of the elevations in the opposite halves sity of the su. 

 of each semicircle is precisely the same in an oblate as in P^'^*^'^^ [*^J|^^ 

 an oblong spheroid of equal ellipticity : so that the ellipti- 

 city must here also be to the disturbing force, where it is 



3 

 greatest, as 1 to 1 — — -, or to the centrifugal force at the 



equator as 1 to 2 — ~. Thus, the centrifugal force being 

 ■^\-^i if the density were uniform, the ellipticity would be 

 ^ix ; but since it is in reality about -j-f-j-, 2 — — — m, and 



n — 1-32, 



